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  • STA256 Lecture 15

Conditional Distribution

• Let $(x_1,x_2)$ be a bivariate Random Vector with a joint distribution $f(x_1,x_2)$ over a valid support.
• The conditional distribution of $x_1$ given $x_2$ is the probability distribution of $x_1$ at a known (fixed) value of $x_2$ from the support of $x_2$
• With Univariate:
  • Conditional Probability
  • $P(A|B)=\frac{P(A\cap B)}{P(B)}$
• The conditional distribution of $x_1$ given $x_2$ is:
  • $f(x_1|x_2)={\displaystyle \frac{f(x_1,x_2)}{\underset{\begin{array}{c}\text{Marginal distibution}\\ \text{of }{x}_2\end{array}}{\underbrace{f(x_2)}}}}$

Discrete Random Vector

• Let $(X_1,X_2)$ be a Discrete Random Vector with a Joint Probability Mass Function:
  • $P(X_1,X_2)$
• The Conditional Distribution of $X_1$ given $X_2$ is:
  • $P(X_1|X_2)={\displaystyle \frac{P(X_1,X_2)}{\underset{\begin{array}{c}\text{Sum the joint}\\ \text{over }{X}_1\end{array}}{\underbrace{P(X_2)}}}}$
  • $\begin{array}{cccc}& & x_2& \\ & \times & 1& 2\\ x_1& 1& 0.3& 0.2& \sum =0.5\\ & 2& 0.1& 0.4& \sum =0.5\\ & & \sum =0.4& \sum =0.6& \sum =1\end{array}$
  • Find the Conditional Distribution of $X_1$ given $X_2$
  • Find the Marginal of $X_2$
    • $\begin{array}{ccc}{X}_2=x_2& 1& 2\\ P(X_2=x_2)& 0.4& 0.6\end{array}$
      • Valid as it adds up to $1$
  • Find the conditional of $X_1$ at all values of $X_2$
  • $\{\begin{array}{l}\underset{―}{X_2=1}\\ P(X_1=1|X_2=1)& =\frac{P(X_1=1,X_2=1)}{P(X_2=1)}\\ & =\frac{0.3}{0.4}\\ & =0.75\\ P(X_1=2|X_2=1)& =\frac{P(X_1=2,X_2=1)}{P(X_2=1)}\\ & =\frac{0.1}{0.4}\\ & =0.25\\ \underset{―}{X_2=2}\\ P(X_1=1|X_2=2)& =\frac{P(X_1=1,X_2=2)}{P(X_2=2)}\\ & =\frac{0.2}{0.6}\\ & =\frac{1}{3}\\ P(X_2=1|X_2=2)& =\frac{P(X_2=1,X_2=2)}{P(X_2=2)}\\ & =\frac{0.4}{0.6}\\ & =\frac{2}{3}\end{array}$
  • Can list all these conditional probabilities.
  • Another way to give the answer:
    • $\begin{array}{ccc}& & P(X_1|X_2=1)\\ x_1& 1& 0.75\\ & 2& 0.25\\ & & \sum =1\end{array}$
    • This is valid, sum is $1$
    • $\begin{array}{ccc}& & P(X_1|X_2=2)\\ x_1& 1& \frac{1}{3}\\ & 2& \frac{2}{3}\\ & & \sum =1\end{array}$
  • Conditional Mean and Variance